The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

TLDR: This review describes mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain.

Abstract: Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain. Newtonian body--limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning, and learning. We outline relevant background material from biomechanics and neurobiology, explain key properties of the hybrid dynamical systems that underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble "peg-leg walker" to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. This final integration in a tractable and illuminating model is an outstanding challenge.

Figures

Fig. 32 Bifurcation set for the hexapedal model in design speed-leg cycle frequency space, showing boundaries of the region in which stable gaits exist, and the speed-frequency protocol adopted. Unstable regions are shaded. The fixed frequency protocol (dashed) encounters instability at low and/or high Vdes, while the piecewise-linear protocol (solid) remains in the stable region, albeit grazing the stability boundary at its break-point. Stability is further improved by adjusting foot placements above Vdes = 0.25, as shown by modified dash-dotted upper boundary. From Figure 14 of [302], with experimental data of [317, Figure 2] indicated by dots. Reprinted with kind permission from Springer Science and Business Media.

Table 1 Acronyms commonly used in this article.

Fig. 1 Templates and anchors: a preferred posture leads to collapse of dimension. The SLIP is shown at left, and a multilegged and jointed model at right. Both share the mass center dynamics of the insect (top). Reprinted from [208] with permission from Elsevier.

Fig. 19 Families of periodic gaits for the fixed and moving COP models with “standard” parameters characteristic of Blaberus discoidalis. From top to bottom the panels show COM velocity vector direction δ, body angular velocity ω ≡ θ̇, or body orientation θ at touchdown, and eigenvalue magnitudes |λ|. (For fixed COP θ = const. at touchdown; for moving COP ω = 0 at touchdown; hence our display of ω and θ, respectively.) Stable branches shown solid, unstable branches dashed (only the neutral and least stable eigenvalues are shown here). Note the saddle-node bifurcations at v̄c, below which periodic gaits do not occur. From Figure 3 of [290]. Reprinted with kind permission from Springer Science and Business Media.

Fig. 7 The schematic two-dimensional space of control architectures. As in Figure 5, single circles represent CPG oscillators, double circles represent mechanical oscillators such as limb components, and triangles represent neural control elements (analogous to operational amplifiers). Reprinted from [208] with permission from Elsevier.

Fig. 21 Partial asymptotic stability. The fixed COP LLS model recovers from an impulsive angular perturbation applied at arrow in (a); evolution of variables at touchdown is shown in (b). The model is neutrally stable in θ and v (corresponding eigenvalues |λ1,2| of unit magnitude) and asymptotically stable in ω (≡ θ̇) and δ. From Figure 2 of [290]. Reprinted with kind permission from Springer Science and Business Media.

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SIAM REVIEW
c

2006 Society for Industrial and Applied Mathematics
Vol. 48, No. 2, pp. 207–304
The Dynamics of Legged Locomotion:
Models, Analyses, and Challenges
∗
Philip Holmes
†
Robert J. Full
‡
Dan Koditschek
§
John Guckenheimer
¶
Abstract. Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace
and economy surpassing our technology. Evolution has shaped the breathtaking abilities of
animals, leaving us the challenge of reconstructing their targets of control and mechanisms
of dexterity. In this review we explore a corner of this fascinating world. We describe
mathematical models for legged animal locomotion, focusing on rapidly running insects
and highlighting past achievements and challenges that remain. Newtonian body–limb
dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical
systems, whose constraints change as legs touch down or lift off. Central pattern gener-
ators and proprioceptive sensing require models of spiking neurons and simplified phase
oscillator descriptions of ensembles of them. A full neuromechanical model of a running an-
imal requires integration of these elements, along with proprioceptive feedback and models
of goal-oriented sensing, planning, and learning. We outline relevant background mate-
rial from biomechanics and neurobiology, explain key properties of the hybrid dynamical
systems that underlie legged locomotion models, and provide numerous examples of such
models, from the simplest, completely soluble “peg-leg walker” to complex neuromuscu-
lar subsystems that are yet to be assembled into models of behaving animals. This final
integration in a tractable and illuminating model is an outstanding challenge.
Key words. animal locomotion, biomechanics, bursting neurons, central pattern generators, control
systems, hybrid dynamical systems, insect locomotion, Lagrangians, motoneurons, mus-
cles, neural networks, periodic gaits, phase oscillators, piecewise holonomic systems, pre-
flexes, reflexes, robotics, sensory systems, stability, templates
AMS subject classifications. 34C15, 34C25, 34C29, 34E10, 70E, 70H, 92B05, 92B20, 92C10, 92C20,
93C10, 93C15, 93C85
DOI. 10.1137/S0036144504445133
1. Introduction. The question of how animals move may seem a simple one.
They push against the world, with legs, fins, tails, wings, or their whole bodies, and
∗
Received by the editors July 12, 2004; accepted for publication (in revised form) April 28, 2005;
published electronically May 2, 2006. This work was supported by DARPA/ONR through grant
N00014-98-1-0747, the DoE through grants DE-FG02-95ER25238 and DE-FG02-93ER25164, and
the NSF through grants DMS-0101208 and EF-0425878.
http://www.siam.org/journals/sirev/48-2/44513.html
†
Department of Mechanical and Aerospace Engineering, and Program in Applied and Computa-
tional Mathematics, Princeton University, Princeton, NJ 08544 (pholmes@math.princeton.edu).
‡
Department of Integrative Biology, University of California, Berkeley, CA 94720 (rjfull@socrates.
berkeley.edu).
§
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia,
PA 19104 (kod@ese.upenn.edu).
¶
Department of Mathematics and Center for Applied Mathematics, Cornell University, Ithaca,
NY 14853 (gucken@cam.cornell.edu).
207

208 HOLMES, FULL, KODITSCHEK, AND GUCKENHEIMER
the rest is Newton’s third and second laws. A little reflection reveals, however, that
locomotion, like other animal behaviors, emerges from complex interactions among
animals’ neural, sensory, and motor systems, their muscle–body dynamics, and their
environments [101]. This has led to three broad approaches to locomotion. Neurobi-
ology emphasizes studies of central pattern generators (CPGs): networks of neurons
in spinal cords of vertebrates and invertebrate thoracic ganglia, capable of generating
muscular activity in the absence of sensory feedback (e.g., [142, 79, 260]). CPGs are
typically studied in preparations isolated in vitro, with sensory inputs and higher
brain “commands” removed [79, 163], and sometimes in neonatal animals. A related,
reflex-driven approach concentrates on the role of proprioceptive
1
feedback and inter-
and intralimb coordination in shaping locomotory patterns [258]. Finally, biomechan-
ical studies focus on body–limb environment dynamics (e.g., [8]) and usually ignore
neural detail. No single approach can encompass the whole problem, although each
has amassed vast amounts of data.
We believe that mathematical models, at various levels and complexities, can
play a critical role in synthesizing parts of these data by developing unified neurome-
chanical descriptions of locomotive behavior, and that in this exercise they can guide
the modeling and understanding of other biological systems, as well as bio-inspired
robots. This review introduces the general problem and, taking the specific case of
rapidly running insects, describes models of varying complexity, outlines analyses of
their behavior, compares their predictions with experimental data, and identifies a
number of specific mathematical questions and challenges. We shall see that, while
biomechanical and neurobiological models of varying complexity are individually rel-
atively well developed, their integration remains largely open. The latter part of this
article will therefore move from a description of work done to a prescription of work
that is mostly yet to be done.
Guided by previous experience with both mathematical and physical (robot) mod-
els, we postulate that successful locomotion depends upon a hierarchical family of
control loops. At the lowest end of the neuromechanical hierarchy, we hypothesize
the primacy of mechanical feedback or preflexes
2
: neural clock-excited and tuned
muscles acting through chosen skeletal postures. Here biomechanical models provide
the basic description, and we are able to get quite far using simple models in which
legs are represented as passively sprung, massless links. Acting above and in concert
with this preflexive bottom layer, we hypothesize feedforward muscle activation from
the CPG, and above that, sensory, feedback-driven reflexes that further increase an
animal’s stability and dexterity by suitably adjusting CPG and motoneuron outputs.
Here modeling of neurons, neural circuitry, and muscles is central. At the highest
level, goal-oriented behaviors such as foraging or predator-avoidance employ environ-
mental sensing and operate on a stride-to-stride timescale to “direct” the animal’s
path. More abstract notions of connectionist neural networks and information and
learning theory are appropriate at this level, which is perhaps the least well developed
mathematically.
Some personal history may help to set the scene. This paper, and some of our
recent work on which it draws, has its origins in a remarkable IMA workshop on
1
Proprioceptive: activated by, related to, or being stimuli produced within the organism (as by
movement or tension in its own tissues) [157]; thus: sensing of the body, as opposed to exteroceptive
sensing of the external environment.
2
Brown and Loeb [48, section 3] define a preflex as “the zero-delay, intrinsic response of a neuro-
musculoskeletal system to a perturbation” and they note that they are programmable via preselection
of muscle activation.

DYNAMICS OF LEGGED LOCOMOTION 209
gait patterns and symmetry held in June 1998, which brought together biologists,
engineers, and mathematicians. At that workshop, one of us (RJF) pointed out that
insects can run stably over rough ground at speeds high enough to challenge the abil-
ity of proprioceptive sensing and neural reflexes to respond to perturbations “within
a stride.” Motivated by his group’s experiments on, and modeling of, the cockroach
Blaberus discoidalis [135, 136, 317, 216] and by the suggestion of Brown and Loeb that,
in rapid movements, “detailed” neural feedback (reflexes) might be partially or wholly
replaced by largely mechanical feedback (preflexes) [49, 227, 48], we formulated simple
mechanical models within which such hypotheses could be made precise and tested.
Using these models, examples of which are described in section 5 below, we confirmed
the preflex hypothesis by showing that simple, energetically conservative systems
with passive elastic legs can produce asymptotically stable gaits [291, 292, 290]. This
prompted “controlled impulse” perturbation experiments on rapidly running cock-
roaches [192] that strongly support the preflex hypothesis in Blaberus, as well as our
current development of more realistic multilegged models incorporating actuated mus-
cles. These allow one to study the differences between static and dynamic stability,
and questions such as how hexapedal and quadrupedal runners differ dynamically (see
sections 3.2 and 5.3).
Workshop discussions in which we all took part also inspired the creation of RHex,
a six-legged robot whose unprecedented mobility suggests that engineers can aspire
to achieving the capabilities of such fabulous runners as the humble cockroach [284,
208]. In turn, since we know (more or less) their ingredients, robots can help us
better understand the animals that inspired them. Mathematical models allow us
to translate between biology and engineering, and our ultimate goal is to produce a
model of a “behaving insect” that can also inform the design of novel legged machines.
More specifically, we envisage a range of models, of varying complexity and analytical
tractability, that will allow us to pose and probe, via simulation and physical machine
and animal experimentation, the mechanisms of locomotive control.
Biology is a broad and rich science, collectively producing vast amounts of data
that may seem overwhelming to the modeler. (In [270], Michael Reed provides a
beautifully clear perspective directed to mathematicians in general, sketching some
of the difficulties and opportunities.) Our earlier work has nonetheless convinced us
that simple models, which, in an exercise of creative neglect, ignore or simplify many
of these data, can be invaluable in uncovering basic principles. We call such a model,
containing the smallest number of variables and parameters that exhibits a behavior
of interest, a template [131]. In robotics applications, we hypothesize the template
as an attracting invariant submanifold on which the restricted dynamics takes a form
prescribed to solve the specific task at hand (e.g., [53, 279, 254, 334]). In both robots
and animals, we imagine that templates are composed [205] to solve different tasks
in various ways by a supervisory controller in the central nervous system (CNS). The
spring-loaded inverted pendulum (SLIP), introduced in section 2.2 and described in
more detail in section 4.4, is a classical locomotion template that describes the center
of mass behavior of diverse legged animals [68, 34]. The SLIP represents the animal’s
body as a point mass bouncing along on a single elastic leg that models the action of
the legs supporting each stance phase: muscles, neurons, and sensing are excluded.
(Acronyms such as CNS, CPG, and SLIP are common in biology, and so for the
reader’s convenience we list those used in this review in Table 1.)
Most of the models described below are templates, but we shall develop at least
some ingredients of a more complete and biologically realistic model: an anchor in
the terminology of [131]. A model representing the neural circuitry of a CPG, mo-